Revisiting the magnetic structure of Holmium at high pressure by using neutron diffraction

Low-temperature neutron diffraction experiments at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P= 8$$\end{document}P=8 GPa have been conducted to investigate the magnetic structures of metallic Holmium at high pressures by employing a long d-spacing high-flux diffractometer and a Paris-Edinburgh press cell inside a cryostat. We find that at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P=8$$\end{document}P=8 GPa and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T=5$$\end{document}T=5 K, no nuclear symmetry change is observed, keeping therefore the hexagonal closed packed (hcp) symmetry at high pressure. Our neutron diffraction data confirm that the ferromagnetic state does not exist. The magnetic structure corresponding to the helimagnetic order, which survives down to 5 K, is fully described by the magnetic superspace group formalism. These results are consistent with those previously published using magnetization experiments.

Magnetism of itinerant electrons 1 have played an important role in condensed matter physics to explain the properties of ferromagnetic metals. In particular, in the 3d transition metals based on Fe, Co, and Ni, the mechanism responsible for their ferromagnetism can be understood within the Stoner model 2 .
On the other hand, the ferromagnetism in the 4f-electron lanthanide metals, such as Gd, Tb, Dy, Ho, Er, and Tm, is explained by the Ruderman-Kasuya-Kittle-Yosida (RKKY) interaction between localized moments of the 4f-electrons mediated by the conduction electrons [3][4][5] . The spatially damped oscillation of the spin polarization of the conduction electrons is responsible of the competition between the ferromagnetic (FM) and antiferromagnetic (AFM) interactions, which often results in an incommensurate helimagnetic structure (HM).
The RKKY interaction constant ( J RKKY ) is highly sensible to subtle lattice contractions that happen at low temperatures. This could destabilize the HM state giving place to a FM ground state. Hereafter, the magnetic transition temperatures between the FM and HM states and between the HM and the paramagnetic (PM) states are denoted as T C and T N , respectively.
The first neutron scattering experiment in this metal was performed at AP by Koehler et al. 28 . They reported that Ho magnetic moments form a basal plane helix below T N = 133 K, and a conical configuration, with a net magnetic moment parallel to the c-axis, below T C = 20 K, in agreement with later neutron experiments 29,30 . In year 1968, Umebayashi et al. 21 studied Tb and Ho at pressures below 1 GPa and temperatures above 80 K, where the pressure dependences of T N and the helical turn angle were measured. It was found that the HM ordering is shifted to lower temperatures when the pressure is increased. In 1988, Achiwa et al. 31 studied the Ho metal up to 2.1 GPa in the temperature range 10 K to T N . The helical pitch angle evolution with temperature was found in agreement with Ref. 21 Thomas et al. 14 performed neutron diffraction experiments at maximum pressures of 6.6 GPa at 89, 110 and 300 K. They established the incommensurate nature of the HM phase and determined the decrease of T N from approximately 122 K at AP at a rate of -4.9 K/GPa up to a pressure of 9 GPa, above which the PM to HM transition vanishes, in agreement with Ref. 21,31 . In 2020, Perreault et al. 25 performed neutron diffraction experiments at maximum pressures of 20 GPa and temperatures above of 10 K. They observed two magnetic transitions below 10 GPa: one to an incommensurate HM phase and another to a conical FM phase. For pressures above 10 GPa in the Sm-type phase, and above 19 GPa in the dhcp phase, the appearance of a magnetic peak at 3 Å and the increase of the intensity of some nuclear peaks were assigned to the presence of a FM ordering below 30 K.
The P dependence of T N and T C , evaluated with magnetic susceptibility 11 (black-empty symbols), a.c and d.c magnetization 12,13 (blue and red filled symbols), and neutron diffraction 25 (green-empty symbols) is showed for Ho in the magnetic phase diagram in Fig. 1.
In our recent a.c. magnetization measurements, using a superconducting quantum interference device (SQUID) magnetometer at zero applied magnetic field ( H dc = 0 T), the signal of the FM anomaly was reduced below any detectable level at a pressure between 5.7 and 8.8 GPa 13 . The anomaly at 5.7 GPa suggested a first order phase transition that remained in the range 0.5-11.6 GPa after remeasuring the sample by decreasing the pressure from 11.2 GPa (the results in the sequence after 11.2 GPa are not visible in Fig. 1 but details can be found in ref. 13 ). Thus, residual strain influenced the suppression of the FM order.
On the other hand, in d.c. magnetization measurements at H dc = 0.5 T, which is one-third of the critical field to the saturated state, the FM anomaly was still observed at 7.3 GPa, and a broad hump appeared at 9.2 GPa 12 . It suggested that the FM order becomes unstable above 7.3 GPa and a short range order composed of small grains is developed at 9.2 GPa. In different magnetization measurements, using a SQUID vibrating-coil-magnetometer method, the development of a ferromagnetic magnetization was observed at 8.2 GPa, which disappeared at 12 GPa 13 . All these facts together stressed that the P region between 8 and 12 GPa is a critical region to understand the FM ordering.
Thus, different series of magnetic measurements showed that; (i) the HM transition was observed at pressures up to 12 GPa, while the FM transition seems to be unstable at approximately 8 GPa, and (ii) the ferromagnetic short range ordering could survive until approximately 11 GPa 12,13 .
However, previous neutron diffraction experiments, covering a d spacing between 1.0 and 3.5 Å, reported that the FM ordering survives until at least 20 GPa 25 . Therefore, it seems that a controversy exists between the Figure 1. P dependence of T N and T C for Ho [11][12][13] . The green point at P = 8 GPa and T N = 105 K has been determined in this study. Blue and red symbols correspond to the data obtained from our previous magnetic characterization studies using an SQUID magnetometer, while the black empty triangles and green empty diamonds correspond to the previous a.c. susceptibility and neutron diffraction experiments, respectively. The size of the error bars, in temperature and pressure, is smaller than the size of the symbols employed. The color bar at the bottom part of the figure indicates the different phase transformations happening as the pressure increases at room temperature according to ref. 32 . The present study focuses on the light-blue shaded region.  25 and the macroscopic magnetic characterization of Ho at high pressures 12,13 .
With the aim to elucidate this controversy, in the present study, by performing neutron powder diffraction experiments, but covering a wide d-spacing region ( 1.4 ≤ d ≤ 50 Å), we determined the magnetic structure at 8 GPa, in the temperature range 5 ≤ T ≤ 300 K . The magnetic superspace group (MSSG) formalism [33][34][35] has been employed to classify the symmetry of the magnetic structure.

High pressure neutron diffraction experiments
Polycrystalline sample of metallic natural Ho with high purity (99.999% ) was purchased from Sigma-Aldrich. Especial care was taken to manipulate the sample minimizing the exposure time to air.
Neutron powder diffraction experiments were carried out on the high-flux 2-axis neutron diffractometer D1B of the Institut Laue-Langevin (ILL) in Grenoble, France. This instrument has a MWGC 1D-detector spanning an angular range of 128 • with a definition of 0.1 • . A Radial Oscillating Collimator (ROC) was installed in order to eliminate the spurious signals produced by the sample environment.
Two data acquisitions were taken at AP and room temperature (RT) for a Ho powder sample inside a 6 mm diameter vanadium can, with neutron wavelengths of = 1.28 Å and 2.52 Å which allowed to explore d-spacings, respectively, of 0.7-15.0 Å and 1.4-50 Å.
The data collection at 8 GPa were performed with = 2.52 Å which correspond to the maximum flux configuration of the D1B instrument. For these acquisitions, the powder was placed in a null-scattering TiZr gasket using a deuterated 4:1 ethanol-methanol mix as pressure transmitter medium, (the same that the one employed before by Perreault et al. at 20 GPa 25 ), which is the typical in all the neutron diffraction experiments. Then, it was introduced inside a VX5/180 Paris-Edinburgh (PE) pressure cell 36,37 equipped with SINE-type sintered diamond anvils 38 . A pressure of 0.12 GPa was applied to the PE cell, which for the sample corresponded to 8 GPa, after calibration with a Pb flake placed with the sample. Then the PE cell was cooled using liquid nitrogen and helium from RT to 5 K. At this temperature, a 4.5 hours isotherm acquisition was performed. Then the sample was warmed back to RT in 10 hours and diffractograms were collected every 15 minutes.
It is well known that the 4:1 ethanol-methanol mixture is not the ideal transmitting media at low temperatures but it is a good approach to the hydrostatic behavior at room temperature for pressures below 10 GPa 39 . Furthermore, the fact that we do not detect any remarkable change, with decreasing the temperature, with the sample inside the PE cell, neither in the background of the diffractograms, nor in the width of the Bragg lines (see Fig. 3), suggests that the quality of the pressure at 5 K is good enough in our experimental conditions. Different crystallographic tools were employed for the determination of the crystal and magnetic structures, which include the FullProf Suite 40 , the ISODISTORT Suite 41,42 , and utilities within the Bilbao Crystallographic Server [43][44][45][46] for the symmetry analysis and visualization. The first remarkable fact is the presence of a new high-intensity peak observed at d ∼ 21 Å that was not present in the diffractograms taken at RT. The low part of the Fig. 3 shows a zoom of the small d-spacing region for the diffractograms collected at T = 5 K (middle part) and RT (bottom part) with the sample inside the PE cell in the cryostat. With this complex sample environment, once the sample is inside the gasket in the PE cell, the intensity is greatly suppressed, since for high pressure experiments the quantity of sample irradiated by the neutron beam is much smaller. Additionally, a huge increase of the background and a broad feature around 3.7 Å are observed even with the ROC in front of the detector, mainly due to the large incoherent scattering of hydrogen from the pressure transmitting fluid 47 .

Results and analysis
The diffractogram observed at RT inside the PE cell, after warming from 5 K, reveals the presence of two distinct sets of reflections. The first set corresponds to the Pb flake, where the reflections are indexed by the cubic space group Fm3m (No. 225), with Pb atoms occupying the WP 4a 48 (see purple diamonds label in Fig. 3). By analysing the position of the reflection (2 0 0) , located at d = 2.367(1) Å, we can determine the cell parameter to be a = 4.734(2) Å. This information allows us to derive a volume per atom of V = 26.52(3) Å 3 /atom, which, when considering the equations of state presented in 49,50 , yields a pressure of 8.0(1) GPa. The other set of reflections corresponds to the Ho sample and is indexed by the same hcp structure ( P6 3 /mmc.1 ′ ) with shorter cell parameters ( a = b = 3.4030(7) Å, c = 5.345(2) Å, and R Bragg = 6.07) than the one observed at AP. Therefore, no structural transition to the Sm-type phase has been observed as the pressure was increased from AP to 8 GPa.
However, at 8 GPa and 5 K, several new magnetic satellites appear, marked with red and orange asterisks in Fig. 3, including the one clearly observed at d ∼ 21 Å. These satellites are indexed with an incommensurate propagation vector � k HM = (0 0 δ) with δ = ±0.2536(1) , in units of c * , in agreement with Ref. 25 . The plus or minus sign in the propagation vector indicates a clock-wise or anti clock-wise propagation. The nuclear peaks, marked with black and grey asterisks in Fig. 3, remain unchanged respect to the ones at RT, suggesting that no FM order is present, even at the lowest temperature. The analysis of this new magnetic state will be elucidated in the next subsection. www.nature.com/scientificreports/ Regarding the reflections due to the Pb flake, at 5 K the reflection (2 0 0) can still be observed at d = 2.367(1) Å, which ensures that the pressure was kept constant (within a 0.1 GPa range) during the warming process. Symmetry analysis. We use the symmetry analysis to reduce the number of possible magnetic ground states in Ho-hcp compatible with the observed propagation vector k HM . We can decompose the magnetic representation for the Ho atom, located at WP 2c, as a direct sum of Irreducible Representations (irreps) of the parent group P6 3 /mmc.1 ′ for the point, (0 0 δ) , of the Brillouin zone (BZ) (we adopted the international notation for the irreps labels and MSSG established in [41][42][43] ), as follows: The basis vectors of each irrep are given in Table 1. The magnetic structure described by both, the 1d-irreps m 2 and m 3 consists of a sinusoidal modulation along the c-axis. Meanwhile, the 2d-irreps, m 5 and m 6 , describe helices in which the magnetic moments are contained in ferromagnetic ab planes, and propagate along the c-axis. However, both models differ greatly when considering the phase-shift between the Ho atoms in the unit cell. While in the case of m 6 the phase-shift is the same as the pitch angle of the helix: φ = 180 × k HM = ±45.65(2) • , in m 5 the phase-shift is given by: φ = 180 × (k HM + 1) = ±225.65(2) • . Therefore, the magnetic structure given by m 6 can be considered as a single helix, while for m 5 the system is composed of two independent helices, one for each atom in the unit cell. The magnetic structure at 5 K, labelled with m 6 , and the φ angle are depicted in Fig. 4.
After a systematic trial and error procedure, it was observed that m 2 , m 3 and m 5 do not fit the data, since they assign zero intensity for the main magnetic peak indexed as (0 0 0) ± � k HM at d ∼ 21 Å. However, the magnetic satellite reflections (red and orange asterisks in Fig. 3) can be correctly fitted by the 2d-irrep m 6 .
The combination between the parent group P6 3 /mmc.1 ′ and the magnetic modulations for Ho atoms given by the irrep m 6 give as a result the P6 3 22.1 ′ (0 0 γ )h00s MSSG. Within the MSSG formalism, the magnetic structure is described by a basic structure, related to the nuclear paramagnetic cell, in addition to a series of magnetic modulation functions that describe the variation from the basic structure of the magnetic moments. In our case, with just one propagation vector k HM and no net magnetic moment (i.e. no existence of a propagation vector � k FM = (0 0 0) ), the magnetic structure is described by: where the internal coordinate x 4 is given by the product of the propagation vector and the position r lj of the atom j in the l-th unit cell ( x 4 = � k HM · � r lj ). The explicit amplitudes of the cosine ( M j,c ) and sine ( M j,s ) components of the magnetic moment for the P6 3 22.1 ′ (0 0 γ )h00s MSSG are given in Table 2. In this MSSG, the Ho atom remains in the WP 2c position and mŴ 2c = 1m� 2 (1) ⊕ 1m� 3 (1) ⊕ 1m� 5 (2) ⊕ 1m� 6 (2). Temperature dependence. Figure 5 shows a 2D plot of the thermo-diffractograms obtained at 8 GPa as the system was heated from 5 K to RT. From them, the onset of the helical magnetic ordering is estimated to occur around T N = 105(2) K, with the appearance of a peak at d ∼ 21 Å ( 2θ ∼ 7 • ), which is in agreement with previous studies [11][12][13]25 .  www.nature.com/scientificreports/ The temperature dependence of the intensity for the (1 0 1) nuclear reflection (in black) and the satellite (0 0 0) ± � k HM (in red) is shown in Fig. 6b. The evolution of these two lines allows to quantitatively distinguish between the magnetic phases that can be present, as the nuclear and satellite peaks chosen should be the most sensible if a FM ( � k FM = (0 0 0) ) or HM order ( k HM ), respectively, should be present. No change is observed neither in the (1 0 1) nuclear reflection, nor other nuclear lines, as the temperature decreases. This fact supports the hypothesis of absence of any FM ordering, or if present, put an upper limit of 0.2µ B to such contribution.
Regarding the evolution with temperature of the satellite (0 0 0) ± � k HM , its intensity was fitted to the power law I ∝ ǫ 2β , where ǫ = T N − T is the reduced temperature (see Fig. 7). The obtained value for the critical exponent is β = 0.40 (1) , which is in good agreement with the value predicted by Bak and Mukamel 51 , and reported by    . 6c).
As the temperature decreases below T N ∼ 105(2) K, the magnetic moment increases until it saturates at 6.94(1)µ B , a value in good agreement with the saturation magnetization obtained at AP and T = 5 K 12 . Meanwhile, the φ angle decreases following two linear dependences from 48.6(1) • at 105 K to 45.65(2) • at 5K, with a change of slope around T ∼ 40 K. This evolution is also coherent with previous neutron diffraction experiments 25 , and can be understood if we consider the strong dependence of the helix period with the axial ratio c/a of the hexagonal phase 8 . Such dependence can also explain the change observed around 40 K, since the ratio c/a is stabilized below that temperature (see Fig. 6a).

Discussion
In our last study using SQUID magnetization measurements it was reported that the HM order survives up to P = 12 GPa, while the disappearance of the FM order, or, at least, a remarkable suppression of ferromagnetic net magnetic moments, was observed above 8 GPa 13 . As the intensity of the FM anomaly could not be detected just above the critical pressure for the phase boundary between the hcp and Sm-type phases, it was suggested that the disappearance of the FM order is related to the structural phase transition.
However, the present neutron diffraction experiments, focused on the magnetic structure at P = 8 GPa, confirm that the hcp structure remains stable until at least 8 GPa. Additionally, an incommensurate HM order is observed below T N = 105(2) K, which persists down to the lowest temperature measured ( T = 5 K), while no evidence of FM ordering was observed within the entire temperature range. Therefore, the present results help to clarify that the FM order becomes unstable and disappear just before the structural phase transition. Furthermore, the magnetic structure observed at 8 GPa is consistent with previous magnetic measurements, taking into account the pressure distribution with the order of ±0.5 GPa during the high-pressure experiment.
In the neutron diffraction study by Perreault et al., the FM order was reported to survive in both the Smtype phase at 14.2 GPa and the dhcp phase at 20.2 GPa 25 . This FM transition was marked by the appearance of a magnetic peak at d = 3 Å accompanied by an increase in the intensity of all nuclear peaks. However, in those experiments only a small region of d-spacing (1.0-3.5 Å) was covered, which made it difficult to elucidate the additional existence of HM order at those pressures, since the expected most intense magnetic signal should appear at larger d-spacing (see Fig. 3). Therefore, it would be interesting to confirm those results by performing neutron experiments at such pressures in a long d-spacing diffractometer.

Conclusion
We conducted neutron diffraction experiments to investigate the magnetic structures of Ho at P = 8 GPa. Our findings indicate that the nuclear symmetry remains unchanged, preserving the hexagonal close-packed (hcp) symmetry at P = 8 GPa. The helimagnetic order persists down to 5 K, and the analysis of its magnetic structure,